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bodies, is the same before and after the collision. This is what is called the vis viva so much attended to by some of the continental writers. Gravity. The most important of all the accelerated motions is that from gravity; as

the motion of a heavy body falling to the ground. During the first second of
time it falls very nearly 16 feet and one inch, or 193 inches. Its velocity in-
creases regularly with the time: hence the motion of gravity is called an equably
accelerated motion. If the time be represented by the line
Ab, and the velocity at the end of Ab by the line Cb,
then the space passed over is represented by the triangle
Abc. Now as triangles are to each other as the squares
of their homologous sides, it follows that the space passed
over is as the square of Ab, or as the square of the time,
which is the great characteristic of equably accelerated
motions. The space is also as the square of Bc, or as the
square of the acquired velocity. Hence it is obvious, that
if it had set out at first with the acquired velocity Bc, it
would have passed over double the space Abc. Hence the acquired velocity
in a second is not measured by 16-^ feet, but by 32^ feet. This velocity is
usually represented by the letter g. The doctrine of equably accelerated motions
is explained by the three following theorems, in which $, *o, t, as before, repre-
sent the space, the velocity, and the time; and g = 32£ fefet. 1. v = gt.

2. s = § f. 3. 2gs = v*. From these we may easily deduce every thing

wanted respecting such motions.

When a body is thrown up from the surface of the earth it is uniformly retarded. Hence the same formulas apply to such motions as to uniformly accelerated motions. Let us suppose a body to be thrown up with a velocity c, and let t denote the time. If gravity did not act, in the time t, it would have moved ct: but during that time, by the action of gravity, it would have fallen | f\ To obtain the real height we must subtract this quantity. It will be there

fore ct — | /*.

Tfte short sketch which we have now given contains the elements of dynamics, stript indeed as much as possible of its mathematical dress ,* and we have been under the necessity of passing over those propositions which could not be rendered intelligible without entering into mathematical disquisitions. The first good system of dynamics that appeared was written by D'Alembert; and the best which we have ever seen in the English language is that which the late Professor Robison, of Edinburgh, inserted in the supplement of the third edition of the Encyclopaedia Britannica,, and which appeared in the first volume of his System of Natural Philosophy, a work unfortunately stopped almost at the outsetbv Dr. Robison's death. We shall now notice the most remarkable papers on Papers in the

n i . i m rr«i Transaction*,

dynamical subjects contained in the Philosophical .transactions. JLhey amount in all to 40; but there are seven of these which can scarcely be considered as of any value, and some of the remaining 33 are rather analyses, or reviews of books, than original papers.

1. The first three papers which occur in the Transactions contain the mathe- Coiiiojonof

. . bodies,

matical doctrine of the collision of bodies. This important part of dynamics was ascertained about the same time by Dr. Wallis, Sir Christopher Wren, and Mr, Huygens, who all communicated their theories of the subject to the Royal Society, within a very short space of time, and without any communication with each other. Dr. Wallis was first: his paper was read to the Society on the 29th of November, 1668. Sir Christopher Wren gave in his paper on the 17th of December, of the same year; and Mr. Huygens sent his on the 5th of January, 1669. So that the interval between them was very small.#

9. The next paper deserving notice is a very valuable one by Dr. Halley on gravity and its properties. The law of the fall of heavy bodies was first determined by Galileo, and the subject was still further prosecuted by his pupil Torricelli. But Sir Isaac Newton was the person who brought it to perfection, and first demonstrated some of the propositions contained in our preceding sketch. Dr. Halley's paper was published before the appearance of the Principia, though it is plain that he was acquainted with many of the principles contained in that admirable book. Indeed he mentions it as almost ready for publication. Dr. Halley has fallen into some mistakes in this paper; though as a whole it possesses considerable value. It contains not only the general properties of gravity, but gives also the outlines of the mathematical theory of #

projectiles.f

3. The next paper is by Mr. Hauksbee, and is not strictly speaking dynamical, but consists of a set of experiments to prove the resistance of the air. I mention it here because the resistance of the air has considerable effect in certain dynamical experiments. Hauksbee found that a marble let fall on glass rebounded higher in vacuo than in common air, and higher in common air than in condensed air.J

4. The next paper is by Mr. John Keill, entitled, Of the Laws of Centripetal Forced He begins by demonstrating the following theorem, which he informs us had been communicated to him by Dr. Halley, but discovered by Newton. If a body urged by a centripetal force move in any curve, then in every point of the curve that force will be in a ratio compounded of the direct ratio of the body's distance from the centre of force, and the reciprocal ratio

* Phil. Trans. Vol. III. p. 864, 867; and Vol. IV. p. 925. f PW1. Trans. 1686. Vol. XVI. p. 3. % Phil. Trans. 1705, Vol, XXIV. p. 1946. § Phil. Trans. 1708. Vol. XXVI. p. 174.

of the cube of the perpendicular on the tangent to the same point of the curve, drawn into the radius of curvature of the same point. From this theorem' which he demonstrates, he deduces, in a very luminous manner, the laws of centripetal forces. But the paper is chiefly remarkable for containing the sentence about the discovery of Auctions, which gave so much offence to Leibnitz, and occasioned the famous controversy between the British and German mathematicians.

5. The next paper contains the repetition of the well known experiment of the guinea and feather falling from the top of an exhausted receiver, and reaching the bottom at the same time. Desaguliers, by joining together a number of glass receivers, contrived to make these bodies fall from the height of eight feet; and found, when the air was well exhausted, that both bodies fell to the bottom at the same instant of time.*

6. The Philosophical Transactions contain no less than 12 papers on the celebrated controvery, whether the force of moving bodies be proportional to their velocity, or to the square of their velocity. The authors of these papers are Dr. Pemberton,f Dr. Desaguliers^ Mr. Eames,§ Dr. Samuel Clarke/|| Dr, Jurin,? Dr. Reid,** and Mr. Milner.tf This, being a dispute about a definition, could not possibly be terminated, both parties being in the right, and both in the wrong. Newton's definition is simplest, and best adapted to mechanical philosophy, and therefore in every respect preferable to that of^Leibnitz. Dr. Reid's paper above referred to explains the real nature of the controversy in a very luminous manner; and may be considered as* sufficient to set the controversy for ever at rest.

7. In the next paper which deserves notice, Dr. Deaaguliers demonstrates a property of the balance, which at first view appears paradoxical. If a man be counterpoised in one scale by a weight in the other, if he presses with a stick against the beam he will become heavier than the counterpoise. On the other hand, if the beam in which the man hangs be half way between the extremity of the balance, and the point of suspension of the balance, and if he presses against any part of the beam on the outside of the scale, he will become lighter than the counterpoise.^ We likewise owe to Dr. Desaguliers some statical experiments described in a subsequent volume of the Transactions.^

8. In the next paper Dr. Jurin gives a general theorem respecting the action of springs, and deduces from it a great number of curious corollaries. The paper is not susceptible of abridgment.* The principle which regulates springs or elastic bodies was first pointed out by Dr. Hooke in this short sentence, Ut tensio sic vis, when he applied springs to regulate the motions of watches. The meaning of this sentence is, that the resistance of the spring is always proportional to the space by which it is driven from its natural situation. If a certain force remove it an inch from its natural situation, it will take twice the force to remove it two inches, and thrice the force to remove it three inches, and so on

* Phil. Trans. 1717. Vol. XXX. p. 717. f phI1« Tr»TM- 1722. Vol. XXXII. p. 57.

J Phil. Trans. 1723. Vol. XXXII. p. 269 and 285; and Vol. XXXVIII. p. 143.
§ PhiL Trans. 1726. Vol. XXXIV. p. 183; and Vol. XXXV. p. 343.
|| Phil. Trans. 1728. Vol. XXXV. p. 381.

f PhiL Trans. 1740. Vol. XLI. p. 607; 1745. Vol. XLIII. p. 423; and Vx>L XLIV. p. 103.
** Phil. Trans. 1748. Vol. XLV. p. 505. ff PhiL Trans. 1778. VoL LXVlII. p. 314.

Xt PhiL Trans. 1729. Vol. XXXVI. p. 128. §§ PhiL Trans. 1737. Vol. XL. p. 62.

9. The next paper is a very elaborate one of Mr. Smeaton, entitled, An experimental Examination of the Quantity and Proportion of Mechanical Power necessary to be employed in giving different degrees of velocity to heavy bodies

from a state of rest.j- In this paper, Mr. Smeaton shows by experiment, that what he calls the mechanical power of a body increases as the square of the velocity. But we must carefully distinguish between this mechanical power and the Newtonian term momentum or quantity of motion. The one is measured by its instantaneous action* the other by its action for a certain time. The momentum by its definition is in the compound ratio of the mass of a body and its velocity, and therefore simply as the velocity in a given body; while the mechanical power by its definition is estimated by the mass compounded with the space it has described in acquiring its velocity. Now since the space fallen is as the square of the acquired velocity, it follows that this force must be as the square of the velocity in a given body.

10. The next paper is a very long and elaborate one by Mr. Landen, entitled, A New Theory of the Rotatory Motion of Bodies affected by forces disturbing

such Motion.% This is an excellent paper, like all the others communicated by <*

that very eminent mathematician, and is of considerable importance in several astronomical discussions.

11. The next paper is by Mr. Biigge, Astronomer Royal at Copenhagen, and is entitled, On the Theory of Pile Driving.^ This paper is very clearly written ; but there are some mistakes in his assumptions, which prevents his ultimate conclusions from being quite accurate. His greatest error respects the friction, which he estimates too high, making it (d+b)b instead of (d~}-±b)b.

12. The next paper is entitled, An Livestigation of the Principles of Progressive and Rotatory Mot ion J\ Mr. Vince, the author of this excellent paper has carried the subject a good deal further than had been done by preceding mathematicians. There is also a set of experiments on the collision of bodies by Mr. Smeaton, in a subsequent volume of the Transactions,*L in which that tery ingenious practical engineer falls into some mistakes, from confounding his mechanical force with the Newtonian momentum.

* Phil. Trans. 1744. Vol. XLIII. p. 472. + Phil. Trans. 1776. Vol. LXVI. p. 450.

% Phil. Trans. 1777. Vol. LXVIL p. 266. § Phil. Trans. 1779. Vol. LXIX. p. 120.

J Phil. Trans, 1780. Vol. LXX. p, 516. f Phil. Trans. 1782. Vol. LXXIL p. 337,

13. We shall notice here a valuable paper by Sir George Shuckburgh Evelyn, on an invariable standard of weights and measures, because we do not know where else it can be better placed. Mr. Whitehurst had been employed, like Sir George, in endeavouring to determine an invariable standard of weights and measures, and the method, which occurred to him as the best, was to determine the difference between the length of two pendulums which vibrate with different degrees of velocity. He had constructed a machine for the purpose which Sir George Shuckburgh procured. The result was, that the difference between the length of two pendulums, which vibrate 42 and 84 times in a minute of mean time in the latitude of London at 113 feet above the level of the sea in the temperature of 68, and the barometer at 30 inches, is equal to 5989358 inches of the parliamentary standard. From this all the measures of superficies and capacity are deducible. Agreeably to the aame scale of inches, a cubic inch of pure distilled water, when the barometer is 29*74 inches, and the thermometer at 66°, weighs 252422 parliamentary grains, From which all other weights may be easily deduced.*

CHAP. IV.
OF MECHANICS.

Ptemn°nof ^E *erm Mechanics is applied in different senses, or at least with very different degrees of latitude by different writers, sometimes comprehending almost the whole of natural philosophy, sometimes being confined to the constrained motions of bodies, and sometimes to the mechanical powers and machines. It will answer our purpose at present to take the word in the last sense; though we shall not scruple to class any paper in the Transactions, which happens to treat of the doctrine of constrained motion, if any such occur, under the present chapter.

The mechanical [powers are ususally reduced under five heads; namely, the lever, the wheel and axle, the pulley, the wedge, and the screw. All these mechanical powers are mentioned by Pappus as known before his time, though

* Phil. Trans. 1798. Vol. LXXXVIII. p. 133.

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